No. It merely illustrates the concept is illogical. Travelling back in time is conceptually flawed. Strictly speaking, the atoms that compose your body would have to return to the position they occupied in the past [not to mention the atoms that compose your 'time machine'.] Otherwise, you would violate causality - the atoms composing you and your time machine would literally be in two different places at the same time! That, at best, is a logical impossibility.

No. It merely illustrates the concept is illogical. Travelling back in time is conceptually flawed. Strictly speaking, the atoms that compose your body would have to return to the position they occupied in the past [not to mention the atoms that compose your 'time machine'.] Otherwise, you would violate causality - the atoms composing you and your time machine would literally be in two different places at the same time! That, at best, is a logical impossibility.

That argument doesn't make sense in terms of the spacetime viewpoint--the atoms that compose your body don't have to rearrange themselves into an earlier state, rather they just have to move to the region of spacetime where the earlier versions of themselves still were in that state, so the older version of you can interact with the younger version. Think of a block of solid ice with various 1-dimensional strings embedded in it--if you cross-section this block, you will see a collection of 0-dimensional points (the strings in cross-section) arranged in various positions on a 2-dimensional surface, and if you take pictures of successive cross-sections and arrange them into a movie, you will see the points moving around continuously relative to one another (in terms of this metaphor, the idea that different frames define simultaneity differently means you have a choice of what angle to slice the ice when you make your series of cross-sections). You shouldn't think of time travel as the points returning to precisely the same configuration they had been in at an earlier frame of the movie; instead, you should just imagine one of the strings curving around into a loop within the 3-dimensional block.

Time travel is allowed by general relativity in certain circumstances--the technical term used is "closed timelike curves"--so unless you think GR contains a logical contradiction, there can't be any logical inconsistency in the idea of backwards time travel. Of course, most physicists would probably bet that closed timelike curves, like singularities, will be eliminated by quantum gravity, but it's not clear that they will be yet, and a number of physicists (Kip Thorne, for example) take time travel to be a real possibility (in the sense of being allowed by the laws of physics, not in the sense of something we'd be able to do in the forseeable future).